nLab
Sandbox

Proposition

Let XX be an absolute neighbourhood retract (ANR) and AiXA \xhookrightarrow{i} X a closed subspace-inclusion. Then ii is a Hurewicz cofibration iff AA is itself an ANR.

(Aguilar, Gitler & Prieto 2002, Thm. 4.2.15)

Proposition

Let XX be a paracompact Banach manifold. Then the inclusion AXA \hookrightarrow X of any closed sub-Banach manifolds is a Hurewicz cofibration.

Proof

Being a closed subspace of a paracompact space, AA is itself paracompact (by this Prop.). But paracompact Banach manifolds are absolute neighbourhood retracts (this Prop.) Therefore the statement follows with Prop. .



π *\pi_\ast

Proposition

If f :X Y f_\bullet \,\colon\, X_\bullet \xrightarrow{\;} Y_\bullet is a morphism of simplicial spaces such that

  1. on simplicial sets of connected components it is a Kan fibration;

  2. all component spaces X nX_n, Y nY_n (nn \in \mathbb{N}) are connected or discrete topological space

then the geometric realization of any homotopy pullback-square of f f_\bullet is a homomotopy pullback?-square in topological spaces.

Write TopSpTopSp for the convenient category of compactly generated weak Hausdorff spaces, and TopSp QuTopSp_{Qu} for the classical model structure on topological spaces in its version for compactly generated spaces (this Thm.).

Proposition

If HGGrp(TopSp)H \subset G \,\in\, Grp(TopSp) is a subgroup-inclusion of topological groups such that the corresponding coset space coprojection is a Serre fibration

GFibG/HTopSp Qu G \xrightarrow{ \;\in Fib\; } G/H \;\;\; \in \; TopSp_{Qu}

(for example in that it admits local sections).

Then the quotient of the universal principal space EGE G by the subgroup HH is weak homotopy equivalent to the classifying space BHB H of HH:

(EG)/HBHHo(TopSp Qu). (E G)/H \;\simeq\; B H \;\;\; \in Ho(TopSp_{Qu}) \,.

HEG×GGEG×(G/H)G H \xrightarrow{\;} \frac{E G \times G}{G} \xrightarrow{\;} \frac{ E G \times (G/H) }{G}

Proposition

Transported through the equivalence of Prop. , the canonical group action (see this Prop.) of the Weyl group W G(H)W_G(H) on the HH-fixed locus Fnctr(EG,BΓ) H Fnctr\big(\mathbf{E}G ,\, \mathbf{B}\Gamma\big)^H becomes, on connected components π 0(CrsHom(H,Γ) adΓ)H Grp 1(H,Γ) \pi_0 \big( CrsHom(H,\,\Gamma) \sslash_{\!\!ad} \Gamma \big) \;\; \simeq \;\; H^1_{Grp}(H,\,\Gamma) , the W G(H)W_G(H)-action on the non-abelian group 1-cohomology of HH from Prop. .

Proof

We make explicit use of the functors L,RL, R constructed in the proof of Prop. . Noticing that RR is a section of LL, we need to (1) send a crossed homomorphism up with RR, (2) there act on it with nn, (3) send the result back with LL. The result is the desired induced action.

Explicitly, by the definition of LL in the proof of Prop. , this way a crossed homomorpism ϕ:HΓ\phi \,\colon\, H \to \Gamma is sent by nN G(H)n \in N_G(H) to the assignment

(1)h(L(n(Rϕ)))(h)=α(n)((Rϕ)(n 1,n 1h)). h \,\mapsto\, \Big(L\big( n \cdot (R \phi) \big)\Big)(h) \;=\; \alpha(n) \Big( (R\phi) \big( n^{-1} ,\, n^{-1} \cdot h \big) \Big) \,.

It just remains to evaluate the right hand side.

Notice that the definition of LL is independent of the choice of σ:G/HG\sigma \,\colon\, G/H \xrightarrow{\;} G (?), and that RR (whose definition does depend on this choice ) is a section for each choice. Hence we may choose σ\sigma in a way convenient way for each nn.

Now if nHN G(H)n \in H \subset N_G(H) then its canonical action on the HH-fixed locus is trivial, and also the claimed induced action is trivial, so that in this case there is nothing further to be proven. Therefore we assume now that nn is not in HH, and then we choose σ\sigma such as to pick n 1n^{-1} as the representative in its HH-coset:

σ([n 1])n 1. \sigma\big( \big[n^{-1}\big] \big) \;\coloneqq\; n^{-1} \,.

With this choice, the right hand side of (1) is evaluated as follows, where we repeatedly use that, by definition and choice of σ\sigma, RϕR\phi assigns the neutral element to the morphism n 1en^{-1} \to \mathrm{e} in the pair groupoid:

(L(n(Rϕ)))(h) =α(n)((Rϕ)(n 1,n 1h)) ==α(n)((Rϕ)(e,n 1h)) =α(n 1hnn)((Rϕ)(n 1h 1n,n 1)) =α(n 1hnn)((Rϕ)(n 1h 1n,e)) =α(n)((Rϕ)(e,n 1hn)) =α(n)(ϕ(n 1hn)). \begin{aligned} \Big(L\big( n \cdot (R \phi) \big)\Big)(h) & \;=\; \alpha(n) \Big( (R\phi) \big( n^{-1} ,\, n^{-1} \cdot h \big) \Big) \\ & \;=\; \;=\; \alpha(n) \big( (R\phi)(\mathrm{e},\, n^{-1} \cdot h) \big) \\ & \;=\; \alpha\big(n^{-1} \cdot h \cdot n \cdot n\big) \big( (R\phi)(n^{-1} \cdot h^{-1} \cdot n,\, n^{-1}) \big) \\ & \;=\; \alpha\big(n^{-1} \cdot h \cdot n \cdot n\big) \big( (R\phi)(n^{-1} \cdot h^{-1} \cdot n,\, \mathrm{e}) \big) \\ & \;=\; \alpha(n) \big( (R\phi)(\mathrm{e},\, n^{-1} \cdot h \cdot n) \big) \\ & \;=\; \alpha(n) \big( \phi(n^{-1} \cdot h \cdot n) \big) \mathrlap{\,.} \end{aligned}

This is indeed the claimed formula (?).

Given αGAut Grp(Γ)\alpha \,G\, \xrightarrow{\;} Aut_{Grp}(\Gamma) as above, consider a subgroup HGH \subset G.

Proposition

The set of crossed homomorphisms HΓH \to \Gamma, with respect to the restricted action of HH on Γ\Gamma, carries a group action of the normalizer subgroup N G(H)GN_G(H) \,\subset\, G, given by

N G(H)×CrsHom(H,G) CrsHom(H,G) (n,ϕ) ϕ nα(n)(ϕ(n 1()n)) \array{ N_G(H) \times CrsHom(H,\,G) &\xrightarrow{\;\;}& CrsHom(H,\,G) \\ (n, \phi) &\mapsto& \phi_{n} \mathrlap{ \;\coloneqq\; \alpha(n) \Big( \phi \big( n^{-1} \cdot (-) \cdot n \big) \Big) } }

Moreover, on crossed-conjugation classes of crossed homomorphisms, hence on first non-abelian group cohomology, this action descends to an action of the Weyl group W G(H)N G(H)/HW_G(H) \coloneqq N_G(H)/H:

H Grp 1(H,Γ)W G(H)Act(Sets). H^1_{Grp}(H,\,\Gamma) \;\;\; \in \; W_G(H) Act(Sets) \,.

Proof

For the first statement: It is clear that this is a group action if only ϕ n\phi_n is indeed a crossed homomorphism. This follows by a direct computation:

ϕ n(h 1h 2) =α(n)(ϕ(n 1h 1h 2n)) definition ofϕ n =α(n)(ϕ(n 1h 1nn 1h 2n)) group property ofN G(H)G =α(n)(ϕ(n 1h 1n)α(n 1h 1n)(ϕ(n 1h 2n))) crossed homomorphism property ofϕ =α(n)(ϕ(n 1h 1n))α(h 1n)(ϕ(n 1h 2n)) action property ofα =α(n)(ϕ(n 1h 1n))α(h 1)(α(n)(ϕ(n 1h 2n))) action property ofα =ϕ n(h 1)α(h 1)(ϕ n(h 2)) definition ofϕ n. \begin{array}{lll} \phi_n( h_1 \cdot h_2 ) & \;=\; \alpha(n) \Big( \phi \big( n^{-1} \cdot h_1 \cdot h_2 \cdot n \big) \Big) & \text{definition of}\; \phi_n \\ & \;=\; \alpha(n) \Big( \phi \big( n^{-1} \cdot h_1 \cdot n \cdot n^{-1} \cdot h_2 \cdot n \big) \Big) & \text{group property of}\; N_G(H) \subset G \\ & \;=\; \alpha(n) \Big( \phi \big( n^{-1} \cdot h_1 \cdot n \big) \cdot \alpha(n^{-1} \cdot h_1 \cdot n) \big( \phi( n^{-1} \cdot h_2 \cdot n ) \big) \Big) & \text{crossed homomorphism property of} \; \phi \\ & \;=\; \alpha(n) \Big( \phi \big( n^{-1} \cdot h_1 \cdot n \big) \Big) \cdot \alpha(h_1 \cdot n) \Big( \phi\big( n^{-1} \cdot h_2 \cdot n \big) \Big) & \text{action property of} \; \alpha \\ & \;=\; \alpha(n) \Big( \phi \big( n^{-1} \cdot h_1 \cdot n \big) \Big) \cdot \alpha(h_1) \bigg( \alpha(n) \Big( \phi\big( n^{-1} \cdot h_2 \cdot n \big) \Big) \bigg) & \text{action property of} \; \alpha \\ & \;=\; \phi_n(h_1) \cdot \alpha(h_1) \big( \phi_n(h_2) \big) & \text{definition of}\; \phi_n \mathrlap{\,.} \end{array}

To see that this action descends to group cohomology, we need to show for

ϕ()=γ 1ϕ()α()(γ) \phi'(-) \;=\; \gamma^{-1} \cdot \phi(-) \cdot \alpha(-)(\gamma)

a crossed conjugation, that there exists a crossed conjugation between ϕ n\phi'_n and ϕ n\phi_n. The following direct computation shows that this is given by crossed conjugation with α(n)(γ)\alpha(n)(\gamma):

ϕ n(h) =α(n)(γ 1ϕ(n 1hn)α(n 1hn)(γ)) assumption with definition ofϕ n =α(n)(γ 1)α(n)(ϕ(n 1hn))α(h)(α(n)(γ)) action property ofα =(α(n)(γ)) 1ϕ n(h)α(h)(ϕ(n)(γ)) definition ofϕ n. \begin{array}{lll} \phi'_n(h) & \;=\; \alpha(n) \Big( \gamma^{-1} \cdot \phi\big( n^{-1} \cdot h \cdot n \big) \cdot \alpha \big( n^{-1} \cdot h \cdot n \big) (\gamma) \Big) & \text{assumption with definition of} \; \phi_n \\ & \;=\; \alpha(n) \big( \gamma^{-1} \big) \cdot \alpha(n) \Big( \phi\big( n^{-1} \cdot h \cdot n \big) \Big) \cdot \alpha(h) \Big( \alpha(n)(\gamma) \Big) & \text{action property of} \; \alpha \\ & \;=\; \big( \alpha(n)(\gamma) \big)^{-1} \cdot \phi_n(h) \cdot \alpha(h) \big( \phi(n)(\gamma) \big) & \text{definition of} \; \phi_n \,. \end{array}

To conclude, we need to show that for nHN(H)n \in H \subset N(H) there is a crossed conjugation between ϕ n\phi_n and ϕ\phi. The following direct computation shows that this is given by crossed conjugation with ϕ(n)\phi(n) (which is indeed defined, by the assumption that nHn \in H):

ϕ n(h) =α(n)(ϕ(n 1hn)) definition ofϕ n =α(n)(ϕ(n 1)α(n 1)(ϕ(h)α(h)(ϕ(n)))) crossed homomorphism property ofϕ =α(n)(ϕ(n 1))ϕ(h)α(h)(ϕ(n)) action property ofα =(ϕ(n)) 1ϕ(h)α(h)(ϕ(n)) crossed homomorphism property ofϕ. \begin{array}{lll} \phi_n(h) & \;=\; \alpha(n) \Big( \phi \big( n^{-1} \cdot h \cdot n \big) \Big) & \text{definition of} \; \phi_n \\ & \;=\; \alpha(n) \bigg( \phi(n^{-1}) \cdot \alpha(n^{-1}) \Big( \phi(h) \cdot \alpha(h) \big( \phi(n) \big) \Big) \bigg) & \text{crossed homomorphism property of} \; \phi \\ & \;=\; \alpha(n) \big( \phi(n^{-1}) \big) \cdot \phi(h) \cdot \alpha(h) \big( \phi(n) \big) & \text{action property of} \; \alpha \\ & \;=\; \big( \phi(n) \big)^{-1} \cdot \phi(h) \cdot \alpha(h) \big( \phi(n) \big) & \text{crossed homomorphism property of} \; \phi \mathrlap{\,.} \end{array}


Let ϕ:GΓ\phi \;\colon\; G \xrightarrow{\;} \Gamma be a crossed homomorphism, hence equivalently a plain group homomorphism (ϕ(),()):GΓG(\phi(-),\,(-)) \,\colon\, G \xrightarrow{\;} \Gamma \rtimes G. Say that another crossed homomorphism ϕ\phi' is nearby if it is so as a plain homomorphism (ϕ(),())(\phi'(-),(-)).

Then the above theorem says that there is an element (γ,h)ΓG(\gamma,\,h) \,\in\, \Gamma \rtimes G such that

(ϕ(g),g)=(γ,h) 1(ϕ(g),g)(γ,h). \big( \phi'(g),\,g \big) \;\; = \;\; \big( \gamma, \, h \big)^{-1} \cdot \big( \phi'(g),\,g \big) \cdot \big( \gamma, \, h \big) \,.

In order for such a conjugation to be a crossed conjugation of the original morphism, we need h=eh = \mathrm{e}.

Notice that we know already that hC(G)h \in C(G) is in the center of GG, since the projection of both sides of the equation to GG must be the identity, by construction of crossed homomorphisms.

Hence the further conjugation of the above equation by (e,h 1)\big(\mathrm{e},\,h^{-1}\big) yields:

(α(h 1)(ϕ(g),g))=(e,h)(γ,h) 1(ϕ(g),g)(γ,h)(e,h 1). \Big( \alpha(h^{-1}) \big( \phi'(g),\, g \big) \Big) \;\; = \;\; \big( \mathrm{e},\, h \big) \cdot \big( \gamma, \, h \big)^{-1} \cdot \big( \phi'(g),\,g \big) \cdot \big( \gamma, \, h \big) \cdot \big( \mathrm{e},\, h^{-1} \big) \,.

Therefore, if the action of GG on Γ\Gamma restricts along the inclusion C(G)GC(G) \xhookrightarrow{\;} G to the trivial action, then

(γ,h)(e,h 1)=(γ,e) \big( \gamma, \, h \big) \cdot \big( \mathrm{e},\, h^{-1} \big) \;\; = \;\; \big( \gamma, \, \mathrm{e} \big)

corresponds to a crossed conjugation

ϕ()=γ 1ϕ()α()(γ). \phi'(-) \;=\; \gamma^{-1} \cdot \phi(-) \cdot \alpha(-)(\gamma) \,.

crossed homomorphisms as sliced functors

Proposition

Internal to some ambient category 𝒞\mathcal{C} with finite limits, let

Then the following are equivalent:

  1. PXP \to X is the GG-quotient coprojection;

  2. PXP \to X is an effective epimorphism.



  • GG-fixed-wise contractibility of Maps(EG,EΓ)Maps(\mathbf{E}G,\mathbf{E}\Gamma) follows from GG-equivariant contraction of EΓ\mathbf{E}\Gamma

  • the universal equivariant principal \infty-bundle is *Maps(EG,BΓ)\ast \longrightarrow Maps(\mathbf{E}G, \mathbf{B}\Gamma) and the point is that the base space is pointed, but no longer pointed connected – but the universal bundle is still that point inclusion (meaning that all other fibers are empty, as admissible for a formally principal bundle)



Proof

The first condition is equivalent to

P× XPPX P \times_X P \rightrightarrows P \to X

being a coequalizer, the second to

P×GPX P \times G \rightrightarrows P \to X

being a coequalizer. But the pseudo-principality condition says that we have an isomorphism (the shear map)

P× XPP×G P \times_X P \simeq P \times G

which identifies these two diagrams.

Maps(𝒳,EG)=(Fnctr(𝒳,EG)×Fnctn(X 0,G)Fnctr(𝒳,EG)) Maps \big( \mathcal{X} ,\, \mathbf{E}G \big) \;=\; \Big( Fnctr \big( \mathcal{X} ,\, \mathbf{E}G \big) \times Fnctn \big( X_0 ,\, G \big) \rightrightarrows Fnctr \big( \mathcal{X} ,\, \mathbf{E}G \big) \Big)
Fnctr(𝒳,EG)Fnctr(𝒳,BG)×G π 0(𝒳) Fnctr \big( \mathcal{X} ,\, \mathbf{E}G \big) \;\; \simeq \;\; Fnctr \big( \mathcal{X} ,\, \mathbf{B}G \big) \times G^{\pi_0(\mathcal{X})}

Last revised on September 19, 2021 at 02:54:25. See the history of this page for a list of all contributions to it.